So I’ve thought about this argument a bit more and concluded that you are correct, but also that there’s a potential fix to get around this objection.
I think that it’s quite plausible that an agent will have an understanding of its decision mechanism that a) let’s it know it will take the same action in both counterfactuals b) won’t tell it what action it will take in this counterfactual before it makes the decision.
And in that case, I think it makes sense to conclude that the Omega’s prediction depends on your action such that paying gives you the $10,000 reward.
However, there’s a potential fix in that we can construct a non-symmetrical version of this problem where Omega asks you for $200 instead of $100 in the tails case. Then the fact that you would pay in the heads case and combined with making decisions consistently doesn’t automatically imply that you would pay in the tails case. So I suspect that with this fix you actually would have to consider strategies instead of just making a decision purely based on this branch.
So I’ve thought about this argument a bit more and concluded that you are correct, but also that there’s a potential fix to get around this objection.
I think that it’s quite plausible that an agent will have an understanding of its decision mechanism that a) let’s it know it will take the same action in both counterfactuals b) won’t tell it what action it will take in this counterfactual before it makes the decision.
And in that case, I think it makes sense to conclude that the Omega’s prediction depends on your action such that paying gives you the $10,000 reward.
However, there’s a potential fix in that we can construct a non-symmetrical version of this problem where Omega asks you for $200 instead of $100 in the tails case. Then the fact that you would pay in the heads case and combined with making decisions consistently doesn’t automatically imply that you would pay in the tails case. So I suspect that with this fix you actually would have to consider strategies instead of just making a decision purely based on this branch.